Tokyo Journal of Mathematics

On the $p$-class Tower of a $\textbf{Z}_{\textrm{p}}$-extension


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For a number field $k$ and a prime number $p$, let $k_{\infty}$ be a $\textbf{Z}_{\textrm{p}}$-extension of $k$ and $X_{\infty}(k)$ the Galois group over $k_{\infty}$ of the maximal abelian unramified $p$-extension of $k_{\infty}$. We first give a sufficient condition, bearing on the norm index of units in the layers of $k_{\infty}$, for $X_{\infty}(k)$ to be finite. When the prime $p$ is 2 and $X_{\infty}(k)\simeq \textbf{Z}/2\textbf{Z}\times \textbf{Z}/2\textbf{Z}$, we study the structure of the Galois group of the maximal unramified $p$-extension of $k_{\infty}$, improving on some previous results in the case of quadratic fields.

Article information

Tokyo J. Math., Volume 31, Number 2 (2008), 321-332.

First available in Project Euclid: 5 February 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R23: Iwasawa theory
Secondary: 11R11: Quadratic extensions


MOUHIB, Ali; MOVAHHEDI, Abbas. On the $p$-class Tower of a $\textbf{Z}_{\textrm{p}}$-extension. Tokyo J. Math. 31 (2008), no. 2, 321--332. doi:10.3836/tjm/1233844054.

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